Some New Simple Inequalities Involving Exponential, Trigonometric and Hyperbolic Functions

The student is introduced to numerical calculations at the initial stages of studying mathematics. The student masters the execution of four arithmetic operations (addition, subtraction, multiplication, division) manually. For calculating trigonometric and exponential functions, square and cubic roots, arithmetic calculators or tables are usually used; this also applies to logarithms. At the same time, confident proficiency in mathematics necessarily involves the ability to perform calculations manually, with at least some accuracy (2–3 significant figures), for any task in natural sciences and mathematics within the school curriculum. This competence enables the student not to perceive computer computations or calculator-based calculations as something incomprehensible — like a trick or a "black box," where input data is provided and output results are obtained. It is understood that the authors do not advocate abandoning calculators when solving school tasks. However, the student should not become entirely dependent on this tool. The article demonstrates methods for calculating the values of exponential, logarithmic, and trigonometric functions, extends the set of exact values of trigonometric functions beyond what is known from the school curriculum (for arguments , , , , ), and illustrates methods for finding square and cubic roots of a number.

Euler’s formula \(e^{it} = \cos (t) + i\sin (t)\) points to the intimate link between the exponential and the trigonometric functions. (Physicist Richard Feynman called the formula “the most remarkable formula in mathematics.” [32]) The generalized trigonometric and hyperbolic functions we have explored thus far lead to generalizations of the exponential and logarithmic functions as well. We engage here in a short exploration of these transcendental functions.

The nonlinear Schrodinger equation (NLSE) with the aid of three order dispersion terms is investigated to find the exact solutions via the extended $ (\frac{G'}{G^{2}})$ -expansion method and the first integral method. Many exact traveling wave solutions, such as trigonometric, hyperbolic, rational, soliton and complex function solutions, are characterized with some free parameters of the problem studied. It is corroborated that the proposed techniques are manageable, straightforward and powerful tools to find the exact solutions of nonlinear partial differential equations (PDEs). Some figures are plotted to describe the propagation of traveling wave solutions expressed by the hyperbolic functions, trigonometric functions and rational functions.

CHAPTER XII - INVERSE CIRCULAR FUNCTIONS, HYPERBOLIC FUNCTIONS AND INVERSE HYPERBOLIC FUNCTIONS

(NOTE: Every chapter ends with Questions to Guide Your Review, Practice Exercises, and Additional Exercises.) P. Preliminaries. Real Numbers and the Real Line. Coordinates, Lines, and Increments. Functions. Shifting Graphs. Trigonometric Functions. 1. Limits and Continuity. Rates of Change and Limits. Rules for Finding Limits. Target Values and Formal Definitions of Limits. Extensions of the Limit Concept. Continuity. Tangent Lines. 2. Derivatives. The Derivative of a Function. Differentiation Rules. Rates of Change. Derivatives of Trigonometric Functions. The Chain Rule. Implicit Differentiation and Rational Exponents. Related Rates of Change. 3. Applications of Derivatives. Extreme Values of Functions. The Mean Value Theorem. The First Derivative Test for Local Extreme Values. Graphing with y e and y . Limits as x a a, Asymptotes, and Dominant Terms. Optimization. Linearization and Differentials. Newton's Method. 4. Integration. Indefinite Integrals. Differential Equations, Initial Value Problems, and Mathematical Modeling. Integration by Substitution--Running the Chain Rule Backward. Estimating with Finite Sums. Riemann Sums and Definite Integrals. Properties, Area, and the Mean Value Theorem. Substitution in Definite Integrals. Numerical Integration. 5. Applications of Integrals. Areas Between Curves. Finding Volumes by Slicing. Volumes of Solids of Revolution--Disks and Washers. Cylindrical Shells. Lengths of Plan Curves. Areas of Surfaces of Revolution. Moments and Centers of Mass. Work. Fluid Pressures and Forces. The Basic Pattern and Other Modeling Applications. 6. Transcendental Functions. Inverse Functions and Their Derivatives. Natural Logarithms. The Exponential Function. ax and logax. Growth and Decay. L'Hopital's Rule. Relative Rates of Growth. Inverse Trigonomic Functions. Derivatives of Inverse Trigonometric Functions Integrals. Hyperbolic Functions. First Order Differential Equations. Euler's Numerical Method Slope Fields. 7. Techniques of Integration. Basic Integration Formulas. Integration by Parts. Partial Fractions. Trigonometric Substitutions. Integral Tables and CAS. Improper Integrals. 8. Infinite Series. Limits of Sequences of Numbers. Theorems for Calculating Limits of Sequences. Infinite Series. The Integral Test for Series of Nonnegative Terms. Comparison Tests for Series of Nonnegative Terms. The Ratio and Root Tests for Series of Nonnegative Terms. Alternating Series, Absolute and Conditional Convergence. Power Series. Taylor and Maclaurin Series. Convergence of Taylor Series Error Estimates. Applications of Power Series. 9. Conic Sections, Parametrized Curves, and Polar Coordinates. Conic Sections and Quadratic Equations. Classifying Conic Sections by Eccentricity. Quadratic Equations and Rotations. Parametrizations of Plan Curves. Calculus with Parametrized Curves. Polar Coordinates. Graphing in Polar Coordinates. Polar Equations for Conic Sections. Integration in Polar Coordinates. 10. Vectors and Analytic Geometry in Space. Vectors in the Plane. Cartesian (Rectangular) Coordinates and Vectors in Space. Dot Products. Cross Products. Lines and Planes in Space. Cylinders and Quadric Surfaces. Cylindrical and Spherical Coordinates. 11. Vector-Valued Functions and Motion in Space. Vector-Valued Functions and Space Curves. Modeling Projectile Motion. Arc Length and the Unit Tangent Vector T. Curvature, Torison, and the TNB Frame. Planetary Motion and Satellites. 12. Multivariable Functions and Partial Derivatives. Functions of Several Variables. Limits and Continuity. Partial Derivatives. Differentiability, Linearization, and Differentials. The Chain Rule. Partial Derivatives with Constrained Variables. Directional Derivatives, Gradient Vectors, and Tangent Planes. Extreme Values and Saddle Points. Lagrange Multipliers. Taylor's Formula. 13. Multiple Integrals. Double Integrals. Areas, Moments, and Centers of Mass. Double Integrals in Polar Form. Triple Integrals in Rectangular Coordinates. Masses and Moments in Three Dimensions. Triple Integrals in Cylindrical and Spherical Coordinates. Substitutions in Multiple Integrals. 14. Integration in Vector Fields. Line Integrals. Vector Fields, Work, Circulation, and Flux. Path Independence, Potential Functions, and Conservative Fields. Green's Theorem in the Plane. Surface Area and Surface Integrals. Parametrized Surfaces. Stokes's Theorem. The Divergence Theorem and a Unified Theory. Appendices. Mathematical Induction. Proofs of Limit Theorems in Section 1.2. Complex Numbers. Simpson's One-Third Rule. Cauchy's Mean Value Theorem and the Stronger Form of L'Hopital's Rule. Limits that Arise Frequently. The Distributive Law for Vector Cross Products. Determinants and Cramer's Rule. Euler's Theorem and the Increment Theorem.

We present a new approach to exponential functions on time scales and to timescale analogues of ordinary differential equations. We describe in detail the Cayley-exponential function and associated trigonometric and hyperbolic functions. We show that the Cayley-exponential is related to implicit midpoint and trapezoidal rules, similarly as delta and nabla exponential functions are related to Euler numerical schemes. Extending these results on any Pad\'e approximants, we obtain Pad\'e-exponential functions. Moreover, the exact exponential function on time scales is defined. Finally, we present applications of the Cayley-exponential function in the q-calculus and suggest a general approach to dynamic systems on Lie groups.

CORDIC algorithms are used for low cost hardware implementation to calculate elementary functions like trigonometric functions, hyperbolic functions, square root etc. This paper presents an area-efficient multiplier-less architecture for computation of exponential and hyperbolic functions. It has been implemented on Virtex-4 FPGA kit and Altera Quartus -II kit. The proposed architecture is suitable for applications that require high frequency as its frequency of operation is higher than the existing architectures reported in literature.

A bootstrap test for the comparison of nonlinear time series

ABSTRACTBy employing the Hirota bilinear (HB) method and starting from the solution with arbitrary functions, we independently analyze the relationship between abundant wave behavior and the superposition formulas of solutions for a (3+1)‐dimensional Boiti–Leon–Manna–Pempinelli (BLMP) equation. Concurrently, we correlate findings with the behaviors of oceanic waves, yielding some valuable insights. By taking the arbitrary functions as exponential functions, polynomial functions, hyperbolic cosine functions, trigonometric functions, and mixed function, numerous solutions to the equation are obtained as follows: the rogue wave solution, the multikink wave solution, the periodic wave solution, and interaction solutions. Through asymptotic analysis, the superposition of exponential functions leads to the increase in the kink wave numbers. For the rogue wave as the rational function solution, reformulating parameter relations using vectors facilitates the precise determination of its emergence time. Additionally, the impacts of parameters and the order of the rational function on the rogue‐wave shape are independently analyzed. Finally, we examine amplitude variations in rogue and periodic waves under the gradual superposition of kink waves. All types of wave profiles are visually displayed in the form of graphics by selecting appropriate parameters.

A modified asymmetric generalized Prandtl–Ishlinskii model for characterizing the irregular asymmetric hysteresis of self-made pneumatic muscle actuators

This is the second of a series of four volumes covering all stages of development of the Calculus, from the last years at school to degree standard. The books are written for students of science and engineering as well as for specialist mathematicians, and are designed to bridge the gap between the works used in schools and more advanced studies. with their emphasis on rigour. This treatment of algebraic and trigonometric functions is here developed to cover logarithmic, exponential and hyperbolic functions and the expansion of all these functions as power series. There is a chapter on curves and the idea of complex numbers is introduced for the first time. In the two final chapters, the author begins a systematic treatment of methods of integrating functions, introducing principles into what often seems rather a haphazard process. This volume, like the others, is well endowed with examples.

The basic principle of this study is to obtain various solutions to the (1+1) dimensional Mikhailov-Novikov-Wang integrable equation (MNWIE). For this purpose, the generalized exponential rational function method (GERFM) is applied to this equation. Thus, several trigonometric functions, hyperbolic functions, and dark soliton solutions to the studied equation are acquired. In this way, some new solutions to the equation that have not been presented before have been obtained. In addition, 2D and 3D graphics of the acquired solutions are drawn for specific values. The obtained results and the graphic drawings of the results have been provided by using Wolfram Mathematica 12.

Constitutive behavior and processing maps of low-expansion GH909 superalloy

Background/Objectives: Integral transforms and iterative techniques play a central role in the theory and solution of integral equations. In this work, generalized special functions, namely Prefunctions and Extended Prefunctions are used to obtain exact and approximate solutions of Volterra integral equations of the second kind. These functions are parameterized by 𝛼 and extend the analytic structure of exponential, hyperbolic and trigonometric functions. These functions are effective tools in solving differential and integral equations. Methods: In this study successive approximations method is used to derive closed-form solutions for two classes of Volterra integral equations with different kernels. For the kernel 𝐾(𝑥, 𝑡) = 𝑥 − 𝑡 with 𝑓 (𝑥, 𝛼) = 𝑝𝑠𝑖𝑛(𝑥, 𝛼) the iterative sequence yields a generalized solution expressed as a series of shifted Prefunctions, reducing to 𝑦(𝑥, 0) = 1 2 (𝑝𝑠𝑖𝑛ℎ(𝑥) + 𝑠𝑖𝑛(𝑥)) for 𝛼 =0 confirming the consistency of the method with known classical results. For the convolution kernel 𝐾(𝑥, 𝑡) = 𝑒−(𝑥−𝑠) with 𝑓(𝑥, 𝛼) = 𝑝𝑠𝑖𝑛ℎ(𝑥, 𝛼) the Picard iteration process leads to the compact representation 𝑦(𝑥, 𝛼) = 𝑝𝑠𝑖𝑛ℎ(𝑥, 𝛼) + 𝑠𝑖𝑛(𝑥, 𝛼 + 1). Findings: Numerical illustrations and graphical results confirm uniform convergence of the successive approximations to the exact solution for various values of 𝛼. For a large value of 𝛼 convergence rate is higher. Novelty: Prefunction -based methods provide generalized frameworks for solving Volterra integral equations and open further directions for applications in mathematical physics and engineering models. 2020 Mathematics Subject Classification: 45Dxx, 34A12. Keywords: Integral Equations; Prefunctions; Successive Approximation Method; Convergence; Picard Iterates

In this paper, two design procedures for sigmoid function using exponential function have been proposed implementing division operation and feedback methodology. Hyperbolic tangent function has also been synthesized using the later. All the proposed circuits along with their building blocks have been implemented using current mode device Operational Transconductance Amplifier (OTA). Sensitivities of the circuit and process parameters on the output have been calculated. Performances of all synthesized circuits have been verified with SPICE simulation, establishing the effectiveness of the proposed methodology.